July 31, 190S] 



SCIENCE 



131 



diwm of expressing these i-elations is of 

 seeondaiy consequenee. 



The matheni-dtician evolves the equation 

 for a paxabola and finds a convenient 

 illustration in the law of projectiles. The 

 engineer finds that a physical result fol- 

 lows from the application of certain forces, 

 and uses the formula merely as a con- 

 venient method of expressing the law. The 

 analogue in the case of mechanical tools is 

 found by regarding a set of drawing in- 

 struments or a, transit or a lathe, as some- 

 thing intelligently designed, properly pro- 

 portioned, accurately made and finely 

 finished, the merit of which lies in its own 

 inherent excellence; or, on the other hand, 

 by considering them as tools adapted for 

 doing a certain range and character of 

 work with a sufficient degree of accuracy 

 and at low cost. 



A manual-training school gives familiar- 

 ity with mechanical tools and mathe- 

 matical study gives familiarity with in- 

 tellectual tools. In work with the manual 

 tool the boy uses it for making something 

 —he learns the principle on which it 

 operates and the way to use it, by making 

 something; if it is something useful it 

 awakens a higher interest than docs some 

 fancy device. Likewise training of engi- 

 neers in mathematics should be by doing 

 something, by the solving of problems, by 

 dealing with real rather than abstract con- 

 ditions. Let this training be secured while 

 applying mathematics to its normal and 

 legitimate purpose as an auxiliary in the 

 study of other branches. 



In the teaching of mathematics for its 

 own sake stress is apt to be laid upon the 

 processes of deriving results rather than 

 the real meaning of the results themselves. 

 An engineer who uses logarithms has no 

 more concern regarding their derivation 

 than the ordinary user of the dictionary 

 for finding the pronunciation of words has 

 in their etymological derivation. The 



ability to re|)r(Hluce demonstrations in 

 higher mathematics from memory with the 

 book shut is often not as important as it 

 is to understand them with tlie book open. 

 In general an engineer, who has occasion to 

 use higher mathematics, will not be inter- 

 ested in evolving difficult equations, nor 

 will he appeal to his memory, but with 

 text-book or reference before him he will 

 seek the things he wants to use. lie should 

 know where to find them and how to use 

 them. 



In emphasizing what a skilled mechanic 

 can make with very ordinary tools, or the 

 true engineer can accomplish with the 

 parallelogram of forces and the rule of 

 three, there is no intention of discrediting 

 the value of fine equipments, either me- 

 chanical or mathematical, if there be the 

 ability to use them. 



Possibly the practical utility of mathe- 

 matics may appear to be urged too 

 strongly, particularly as the writer really 

 believes in thorough mathematical train- 

 ing, but he has seen so many cases in which 

 mathematical instruction has never been 

 digested and assimilated, he has seen simple 

 problems confused by unnecessary mathe- 

 matical complications, he has seen men 

 satisfied with results which ai-e absurd 

 because of some mathematical equations— 

 sometimes quite unnecessary— which seem 

 to obliterate common-sense perspective, 

 and he recalls the new insight into mathe- 

 matics which came through "Analytic 

 Mechanics" under Professor S. W. Robin- 

 son at the Ohio State Univei-sity, and 

 "Problems in Mechanics," under Dr. 

 Fabian Franklin at Johns Hopkins 

 University, that he feels there is little 

 danger in over-emphasizing the importance 

 of concrete training in mathematical 

 study.^ 



' Both of these teachers of mathematics liad 

 been trained as engineers and liad practised tlio 

 profession. 



